How do I make sure my objects are always touching the grid floor?
There are different ways. If your object has a bottom face that is supposed to touch the plane on z = 0, you select that face in edit mode, call shift+s, cursor to selection. 
Then switch to object mode and change your object's origin by selecting origin to 3D cursor, as shown here:



Then, still in object mode, you can move your object to z = 0 by entering the value directly into the coordinates box.